The unsteady nature of a DSW meant that the study of these wave forms fell behind that of solitary waves (solitons).
The breakthrough in their study came from the development of Whitham modulation theory [1]}, [2]}, [3]}, also
termed the method of averaged Lagrangians. Whitham modulation theory is a powerful method to analyse slowly varying linear,
and importantly, nonlinear, periodic dispersive waves. It is related to the general method of multiple scales [1]}.
Whitham modulation theory generates so-called modulation equations for the parameters , such as amplitude, wavenumber and mean
height, describing a modulated (slowly) varying wavetrain. If these modulation equations are hyperbolic, the underlying nonlinear
periodic wave is stable and if they are elliptic, the wave is unstable. Whitham used modulation theory to give a theoretical
explanation of the Benjamin-Feir instability of gravity water waves [1]}, [6]}, showing that the modulation
equations for these waves are elliptic if the product \(kh\)  of the depth of the water \(h\)  and the wavenumber \(k\)  is above a
critical value. The breakthrough for the theory of DSWs was the realisation that if the modulation equations for a nonlinear,
dispersive wave equation form a hyperbolic system, then a simple wave solution of these equations corresponds to a DSW, the
first DSW solution being that for the KdV equation [7]} based on its previously derived modulation equations
[1]}, [3]}. The original derivation of the KdV modulation equations [1]}, [3]} relied on extensive
manipulation of what are essentially elliptic integrals as the periodic wave solution of the KdV equation is given
in terms of the Jacobian elliptic cosine function \(\mathop {\rm cn}\nolimits z\) . The KdV equation is the standard example of an equation which is
exactly integrable via the method of inverse scattering [1]}, [13]}. Using techniques from functional analysis,
it was found that this spectral solution could be used to immediately determine the modulation equations for the KdV equation
[14]}, with the methods applicable to any integrable nonlinear, dispersive wave equation.
